Answer
$x=5-\sqrt{\frac{51}{2}}$, $ x= 5+\sqrt{\frac{51}{2}}$
Work Step by Step
Use the completing the square method. $$
\begin{aligned}
\frac{2}{3}(x-5)^2-7 & =10 \\
\frac{2}{3}(x-5)^2 & =10+7 \\
\frac{2}{3}(x-5)^2 & =17 \\
(x-5)^2 & =17 \cdot \frac{3}{2} \\
(x-5)^2 & =\frac{51}{2}\\
x-5 & = \pm \sqrt{\frac{51}{2}}
\\ x & =5 \pm \sqrt{\frac{51}{2}}.
\end{aligned}
$$ The solutions are:
$$
\begin{aligned}
x & =5-\sqrt{\frac{51}{2}}\approx -0.05 \\
x & =5+\sqrt{\frac{51}{2}}\approx 10.05.
\end{aligned}
$$ Define the following version of the function and plot it. We see that the zeros are where they should be. $$
f(x) = \frac{2}{3}(x-5)^2-17.
$$
